From 966048bd3594eac4d3398992c8ad3143e290303b Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 8 Apr 2021 16:49:46 +0200
Subject: Expand knowledge base, add /sources/

---
 content/know/concept/cauchy-strain-tensor/index.pdc | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'content/know/concept/cauchy-strain-tensor/index.pdc')

diff --git a/content/know/concept/cauchy-strain-tensor/index.pdc b/content/know/concept/cauchy-strain-tensor/index.pdc
index f150723..cb48377 100644
--- a/content/know/concept/cauchy-strain-tensor/index.pdc
+++ b/content/know/concept/cauchy-strain-tensor/index.pdc
@@ -82,7 +82,7 @@ we expand the middle term to first order in $\va{a}$:
 $$\begin{aligned}
     \va{u}(\va{x} + \va{a})
     \approx \va{u}(\va{x}) + a_x \pdv{\va{u}}{x} + a_y \pdv{\va{u}}{y} + a_z \pdv{\va{u}}{z}
-    = \va{u}(\va{x}) + \va{a} \cdot \nabla \va{u}(\va{x})
+    = \va{u}(\va{x}) + (\va{a} \cdot \nabla) \va{u}(\va{x})
 \end{aligned}$$
 
 With this, we can now define the "shift" $\delta\va{a}$
@@ -91,7 +91,7 @@ as the difference between $\va{a}$ and $\va{A}$ like so:
 $$\begin{aligned}
     \delta{\va{a}}
     \equiv \va{a} - \va{A}
-    = \va{a} \cdot \nabla \va{u}(\va{x})
+    = (\va{a} \cdot \nabla) \va{u}(\va{x})
 \end{aligned}$$
 
 In index notation, we write this expression as follows,
@@ -245,8 +245,8 @@ is easy to express using the displacement field $\va{u}$:
 $$\begin{aligned}
     \boxed{
         \delta(\dd{\va{l}})
-        = \dd{\va{l}} \cdot \nabla \va{u}
-        = (\nabla \vec{u})^\top \cdot \dd{\va{l}}
+        = (\dd{\va{l}} \cdot \nabla) \va{u}
+        %= (\nabla \vec{u})^\top \cdot \dd{\va{l}}
     }
 \end{aligned}$$
 
@@ -259,9 +259,9 @@ $$\begin{aligned}
     = \delta(\va{a} \cross \va{b} \cdot \va{c})
     &= \delta\va{a} \cross \va{b} \cdot \va{c} + \va{a} \cross \delta\va{b} \cdot \va{c} + \va{a} \cross \va{b} \cdot \delta\va{c}
     \\
-    &= (\va{a} \cdot \nabla\va{u}) \cross \va{b} \cdot \va{c}
-    + \va{a} \cross (\va{b} \cdot \nabla\va{u}) \cdot \va{c}
-    + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla\va{u})
+    &= (\va{a} \cdot \nabla) \va{u} \cross \va{b} \cdot \va{c}
+    + \va{a} \cross (\va{b} \cdot \nabla )\va{u} \cdot \va{c}
+    + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla) \va{u}
 \end{aligned}$$
 
 We can reorder the factors like so
@@ -303,7 +303,7 @@ $$\begin{aligned}
     \delta(\dd{V})
     = \delta(\va{c} \cdot \dd{\va{S}})
     = \delta\va{c} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
-    = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
+    = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
 \end{aligned}$$
 
 By comparing this to the previous result for $\delta(\dd{V})$,
@@ -311,7 +311,7 @@ we arrive at the following equation:
 
 $$\begin{aligned}
     \nabla \cdot \va{u} (\va{c} \cdot \dd{\va{S}})
-    = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
+    = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
 \end{aligned}$$
 
 Since $\va{c}$ is dot-multiplied at the front of each term,
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